MINIMIZING VOLUMES OF KÄHLER MANIFOLDS OF CODIMENSION ONE

Abstract

Let $M$ be a compact complex Kähler manifold of dimension $n\geq 3$, and suppose that $N\hookrightarrow M$ is a closed complex submanifold of dimension $n-1$. Denote by $\mathcal{KC}(1)$ the space of classes of Kähler forms $[\omega]\in H^{1,1}(M,\mathbb R)$ that define Kähler metrics of volume $1$ on $M$, and define $\mathbf f:\mathcal{KC}(1)\longrightarrow\mathbb R$ by $\mathbf f([\omega])=({1}/{(n-1)!})\int _N\omega^{n-1}$, equal to the volume of $N$ in the metric induced by $\omega$. In this paper, the critical points of $\mathbf f$ are studied. The Riemann–Hodge bilinear relations of $M$ and $N$ are shown to imply that any critical point of $\mathbf f$ is the strict global minimum; also, the hard Lefschetz theorem determines the critical point $[\omega]$. A positive multiple of $[\omega]\in H^2(M,\mathbb R)$ is the Poincaré dual of the homology class of $N$. If this is applied to the Theta divisor of the Jacobian $J(C)$ of a compact Riemann surface $C$, the Theta metric minimizes the volume of the Theta divisor within all Kähler metrics of volume one on $J(C)$.